In the field of asymmetric communication process, the first and most well-known solution was invented in 1977. It was the subject of U.S. Pat. No. 4,405,829 filed by the inventors Rivest, Shamir and Adleman (RSA) on Dec. 14, 1977. RSA solution has two types of keys. The first key (public) allows the enciphering of messages and the second (private) allows their deciphering. This process is the first asymmetric cryptography, whose name comes from the fact that the keys for enciphering process and deciphering process are completely different. In an open network, each member possesses a public key known by anyone and the private key that must never be revealed. The RSA process can also be used for various purpose including signature and authentication, etc.
This cryptographic communication process has serious drawbacks. The numbers to be used are very large, with a currently believed requirement of 1000 bit keys for security. Thus, calculation complexity is high and the signatures are very long. Moreover, the security of RSA would be compromised if new breakthroughs in integer factorization were to be achieved. For instance, should the intense global development efforts of a quantum computer be successful, it has been shown that RSA would be unusable for secure encryption.
Alternative asymmetrical cryptographic communication processes have been sought to replace RSA. One type of new methods is to use maps (functions) of multi-variables. This idea has a very strongly support by a proven result in computation theory that solving a set of general polynomial equations over finite fields is impossible (NP-hard), which therefore excludes the predicted attack method of any future quantum computer.
One of the first suggestions of using multi-function maps is the MATSUMOTO-IMAI algorithm, whose security however is entirely insufficient due to vulnerability to a linearization attack method.
Another suggestion is the Tame Transformation Map (TTM) cryptographic process, U.S. Pat. No. 5,740,250 to T. Moh, filed Aug. 9, 1996, which is based on the so called tame transformations in algebraic geometry (for which we prefer a different mathematical name, the de Jonquieres maps). In the TTM cryptographic process, the main map, namely the cipher, is a composition of 4 maps, two linear maps at each end and two tame transformations in the middle. The distinguishing feature of TTM is that instead of treating one large number, it treats a large number of small numbers. The main construction is the establishment of a special equation, which ensures the security and efficiency of the system. However, due to the rigid constructions, in particular, the use of only low rank degree two polynomials, it has been show that all of its implementable schemes are insecure by either minirank method attack or linearization method attack.
Patarin patented another asymmetrical cryptographic schema, called the Hidden Field Equation method (HFE), a generalization of the Matsumoto-Imai system, which is based on low degree public polynomial equations with values in a finite field K, described in U.S. Pat. No. 5,790,675. The secret key makes it possible to hide polynomial equations with value in extensions of the finite field K, thus enhancing security. However, the main map, namely the cipher, is made of the composition of only linear maps and maps of polynomial of either only one variable of the hidden field equation or with small variables however still using only one field. Thus, the structure depends only on one field. A recent relinearization method and more general methods were constructed to attack the HFE system and some of the HFE systems have been broken. For example, a $500 challenge set by the inventor was actually broken with a PC by a French mathematician Jean-Charles Faugere using the Groebner basis. These attack methods showed that the security can be ensured only if the degree of the low degree public polynomials is not too low. However the higher the degree becomes, the slower and the more complex the decryption process becomes. Though a small variable map is indeed suggested, without the idea of using additional hidden equations, the complexity to invert such a map in this case makes it much less efficient, therefore unusable.
These deficiencies in the known encryption techniques are described in publications such as J. Ding, D. Schmidt. A defect of the TTM implementation schemes, University of Cincinnati, Preprint 2003;    Jacques Patarin, Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt '88, Volume 0963, pp 0248, Lecture Notes in Computer Science, Springer-Verlag Berlin Heidelberg;    Nicolas T. Courtois, The Security of Hidden Field Equations (HFE), Volume 2020, pp 0266, Lecture Notes in Computer Science, Springer-Verlag Berlin Heidelberg;    Aviad Kipnis, Adi Shamir, Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization, Volume 1666, pp 0019, Lecture Notes in Computer Science, Springer-Verlag Berlin Heidelberg;    Nicolas T. Courtois, Jacques Patarin, About the XL Algorithm over $GF(2)$, Volume 2612, pp 141–157, Lecture Notes in Computer Science, Springer-Verlag Berlin Heidelberg; and    Louis Goubin, Nicolas T. Courtois, Cryptanalysis of the TTM Cryptosystem, Volume 1976, pp 0044, Lecture Notes in Computer Science. 
In addition to the previously described asymmetric cryptography, many if not most applications rely upon symmetric cryptography. In the field of symmetric communication process, the earliest inventions trace back to the very beginning of human civilization. The more recent stories are those stories of code breaking in the Second World War. The famous most recent one is the Data Encryption Standard (DES) invented by IBM, a 64-bit implementation that was set as the security standard by the US government. Most recently, a new standard, Advanced Encryption Standard (AES) has accepted.
The need for the new standard with ever increasing key lengths is similar to that described above for asymmetric encryption. Advances in processing capabilities increasing make existing symmetric encryption vulnerable.
With increasing dependence on electronic communication for sensitive transactions, the need for improvements to both symmetric and asymmetric cryptographic systems is becoming pronounced. In particular, a need for enhanced security exists, which is compounded by the need for computational and transmission efficiency. For example, consumers have a wide range of portable electronic devices (e.g., personal digital assistants (PDA), notebook computers, web-enabled wireless telephones, smart cards, etc.) that may be used to effect a financial transaction. However, the available processing capability and network transmission bandwidth may be limited in some instances. Providers of various financial transactions wish to make their services available to a large number of institutions and consumers. Yet these providers also need to maintain a certain degree of security to avoid embarrassing breaches of privacy and monetary damages.
Consequently, a significant need exists for an improved asymmetric and an improved symmetric encryption approach that are secure, even should significant breakthroughs occur in computational speed and capacity.